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The inch is customarily subdivided in dyadic rather than decimal fractions; similarly, the customary divisions of the gallon into half-gallons, quarts, and pints are dyadic. The ancient Egyptians also used dyadic fractions in measurement, with denominators up to 64.[1]

Because they are closed under addition, subtraction, and multiplication, but not division, the dyadic fractions form a subring of the rational numbers Q and an overring of the integers Z. Algebraically, this subring is the localization of the integers Z with respect to the set of powers of two.

The set of all dyadic fractions is dense in the real line: any real number x can be arbitrarily closely approximated by dyadic rationals of the form ⌊2ix⌋/2i{\displaystyle \lfloor 2^{i}x\rfloor /2^{i}}. Compared to other dense subsets of the real line, such as the rational numbers, the dyadic rationals are in some sense a relatively "small" dense set, which is why they sometimes occur in proofs. (See for instance Urysohn's lemma.)

An element of the dyadic solenoid can be represented as an infinite sequence of complex numbers q0, q1, q2, ..., with the properties that each qi lies on the unit circle and that, for all i > 0, qi2 = qi − 1. The group operation on these elements multiplies any two sequences componentwise. Each element of the dyadic solenoid corresponds to a character of the dyadic rationals that maps a/2b to the complex number qba. Conversely, every character χ of the dyadic rationals corresponds to the element of the dyadic solenoid given by qi = χ(1/2i).

The surreal numbers are generated by an iterated construction principle which starts by generating all finite dyadic fractions, and then goes on to create new and strange kinds of infinite, infinitesimal and other numbers.

Time signatures in Western musical notation traditionally consist of dyadic fractions (for example: 2/2, 4/4, 6/8...), although non-dyadic time signatures have been introduced by composers in the twentieth century (for example: 2/., which would literally mean 2/8⁄3, or 9/14). Non-dyadic time signatures are called irrational in musical terminology, but this usage does not correspond to the irrational numbers of mathematics, because they still consist of ratios of integers. Irrational time signatures in the mathematical sense are very rare, but one example (√42/1) appears in Conlon Nancarrow's Studies for Player Piano.

As a data type used by computers, floating-point numbers are often defined as integers multiplied by positive or negative powers of two, and thus all numbers that can be represented for instance by IEEE floating-point datatypes are dyadic rationals. The same is true for the majority of fixed-point datatypes, which also uses powers of two implicitly in the majority of cases.